GAZA CITY, Gaza Strip — Eight Palestinian militants emerged from a tunnel some 300 yards inside Israel on Saturday morning, surprising an Israeli border patrol with a rocket-propelled grenade shot at their jeeps and starting a gunbattle that left two Israeli soldiers and one militant dead before the Palestinians retreated underground, the Israeli military said.

Saturday afternoon, two more militants entered Israel, a military spokesman said, either through a tunnel or breaching the border fence, and carrying tranquilizers and handcuffs. Israeli soldiers fatally shot one; the other was wearing an explosive vest that detonated, killing him.

Hours later, another militant slipped through a different tunnel into Israeli territory and, according to the military, fired on troops who killed him.

The infiltrations came as the Israeli military pressed deeper into Gaza in an intensifying ground war it says is meant to destroy a labyrinth of tunnels leading from Gaza into Israel before they can be used for launching attacks.

The morning clash was the first time in the current war that the militants killed soldiers inside Israel, and came just three days after another infiltration that the military said was the immediate trigger for adding a ground invasion to a deadly air campaign.

The developments also came as Israeli officials revealed that the tunnel network was far more extensive than had been publicly known, with 13 tunnels discovered and an estimated dozens more suspected.

Though the government has said the ground campaign will be limited and aimed only at the tunnels, the growing intensity of the battles that are pushing deeper into civilian areas suggested that the fighting could grow far worse.

Already, tens of thousands have been forced to flee. At one hospital in northern Gaza, the director said that 50 casualties arrived in just three hours Saturday morning, a number that had been typical for an entire day during Israel’s air campaign that began July 8.

The Israelis dropped leaflets urging residents of two crowded refugee camps, Al-Bureij and Al-Maghazi, to evacuate, raising alarms from the United Nations, which said that shelters were already overwhelmed and in danger of running out of supplies.

Among the dead in recent fighting were eight members of a Palestinian family, including four children, killed by an Israeli artillery barrage Friday night. During the funeral Saturday, artillery and small-arms fire echoed nearby from clashes between militants and Israeli forces. Hamas rockets whooshed into the sky from a nearby launch site, and some mourners hurried away before the ceremony was over.

Tunnels under the border with Gaza have had a powerful hold on the Israeli psyche since 2006, when Hamas fighters used one to capture an army lieutenant, Gilad Shalit, who was held for six years. Lieutenant Colonel Peter Lerner, a military spokesman, said a task force had been working for a year on a plan for a ground invasion to destroy the tunnels.

Israeli troops have uncovered 13 tunnels from Gaza into Israel since the start of the ground operation, some of them as much as 30 yards underground, with multiple entry points beneath greenhouses and open fields, he said Saturday. He added that the 13 tunnels, which he now said were under Israeli control, were all over the periphery of Gaza and that he believed there were “tens” more.

The military said the infiltrators Saturday had planned “a lethal attack” in a nearby community, but did not name the town.

Lerner said troops were in the process of demolishing the tunnels, and have been engaged in “urban warfare” inside Gaza.

A rocket fired from Gaza killed an Israeli, Odeh Lafia al-Waj, 32, in a Bedouin village near Dimona, injuring four members of his family, including a 3-month-old girl who was critically hurt. He was the first Israeli civilian killed by one of about 1,600 rockets fired by Hamas, most of which are intercepted by defense systems or fall in open fields. Another was killed by a mortar shell close to Gaza.

The Palestinian death toll in Gaza rose to 301 since July 8, including 72 children, 24 women, and 18 elderly people, with more than 2,000 wounded, the Palestinian health ministry said. About 75 percent of the casualties have been civilians, according to a United Nations count.

As the funeral procession for the Abu Jarad family wound through the streets of Beit Lahiya in northern Gaza, men took turns carrying the bodies of an infant and a toddler, wrapped in bloody white shrouds. The adults’ bodies were wrapped in the yellow flags of Hamas’s rival party, Fatah.

In Khan Younis, in central Gaza, seven people were killed, mostly men, and others were wounded when a drone struck a group of people in the middle of the city, the health ministry said.

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